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Mathematical analysis and simulations



  • In order to go further into the details of mathematical analysis of the FIFO, let us remind you the set of equations used and the corresponding network (note that you can have an explanation on the way they have been built on the Model Construction page):

FliA dynamics.jpg

where CFP, YFP, and RFP will be denoted below as respectively Z1,Z2 and Z3. We have implemented this system using Matlab (see the corresponding codes)

  • Then, in order to obtain simulations of the expected FIFO behaviour, we set ideal conditions, by imposing flhDC as a step function, equal to one then to zero.

FlhDC Test FIFO cresc.jpg
FlhDC Test FIFO decresc.jpg
  • We saw during the overall description of the core system that without FliA, the FIFO would presumably not work. We then simulated a first system, where [FliA] stays to zero value to confirm that qualitative conclusion :

Essai without fliA.jpg

Indeed, we may see that there is a LIFO behavior rather than the FIFO we expect...

  • Then, we simulated the entire system (that means with FliA), to check if we had
    • the lasting burst due to FliA (more important for Z3 than for Z2, and more important for Z2 than Z1) in the increasing phase.
    • the effect of fliA which maintained the concentrations to their maximum (more important for Z3 than for Z2, and more important for Z2 than Z1) in the decreasing phase. We have obtained the following curves, which are in perfect agreement with the theoretical definition of a 'FIFO' process :

Essai with fliA.jpg

  • FliA enables the curves to cross, and adds a delay on the genes that are most affected, with gives a better observability of the FIFO behavior.


  • In addition to the previous FIFO system, we will add a negative feedback on FlhDC promoter to observe oscillations. Thus, we obtained the following network and the corresponding equations (notice that we decide to model in a single arrowthe process through which that EnvZ repress FlhDC):
Core system.jpg

Eqn flhDC.jpg
FliA dynamics.jpg
Eqn EnvZ-RFP.jpg

If you want more details about the way we have built them, you can go to the model construction page.

If one tries to draw simulations of this system, he will obtain this kind of results :

Steady state.jpg

In a nutshell : It does not work !
Nonetheless, there are two questions we could focus on:

  • Is it possible to prove theoretically that it will not produce oscillations ? For detailled answers : mathematical analysis.
  • Does this result depend on the integration methods used ? For more detailled considerations : Impact of integration methods.

Finally, the logical continuation of the process will be to try to improve the biological system in order to find oscillations. This will be described in the part below, as well as in the section System Improvements.

System analysis

Simulations of the core system displayed above reveal that it does not exhibit an oscillatory behavior. In this section we use the model of the core system to try to figure out the contribution of some key characteristics of the network topology on the dynamic of the system. This analysis is done by successively simulating altered forms of the system.

The following table lists the different forms of the system considered and their respective outcome :

Description System Simulation Evaluation Observations
Core system (without modifications)
Core system mini.png
Orgsad.gif Non oscillating system.
Removing the activation of envZ via FlhDC Core system delta flhDC envZ dotted mini.png
Orgstar.gif Non oscillating system. Activation of envZ via FlhDC has a negative effect on the presence of oscillations. This interaction creates a 'shortcut' in the loop, diminishing the effect of Flia in the system. This interaction is already low in the core system, removing it has a moderate positive effect.
Removing the auto-activation of flia Core system delta flia flia.png
Orgstar.gifOrgstar.gif Damped oscillations. Auto-activation of Flia slows its dynamic. For oscillation to appear, different species need to have comparable time scales : oscillations are much more easier without Flia auto-activation.
Removing both: the activation of envZ via FlhDC and the auto-activation of flia Core system delta flhDC envZ delta flia flia.png
Orgstar.gifOrgstar.gif Damped oscillations
Enhancing the inhibition of FlhDC via envZ; Removing both: the activation of envZ
via FlhDC and the auto-activation of flia
Core system envZ plus.png
Orgstar.gifOrgstar.gifOrgstar.gif Damped oscillations with greater amplitude
Synchronized oscillations should be searched taking the last system as a starting point.

This analysis shows that the core system and its altered variants can only provide damped oscillations. In the next section, we examine more significant topology improvements of the core system.


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Mathematical analysis Effects of integration methods